The proposed research is in biostatistical methods associated with event time data (data on survival times, detection times of carcinogenic growths, recurrence times of symptoms, etc.). The objective is to develop additional statistical models for such data, along with associated methods of statistical analysis. The focus will be on semiparametric Bayesian models and methods. The semiparametric nature allows considerable generality and applicability but enough structure for useful physical interpretation and understanding for particular applications in medical research. Recent advances in Bayesian theory and computations make the study of complex models and data structures feasible. Four categories of event time data will be considered: univariate survival data (survival times for a group of unrelated patients), multiple event time data (successive repeated events in each of a group of unrelated patients), multivariate survival data (survival times for a group of patients who are related to each other either genetically or environmentally), two independent events in tandem to every patient (such as infection and onset of disease). Each model considered has been subjected to some kind of censoring mechanism, such as right censoring, grouping (due to inexact measurement or periodic followup), interval censoring (due to missed followups). Monte Carlo algorithms, including data augmentation and Gibbs sampling, will be used to deal with the complexity of the model as well as the censoring or grouping present in the data. The research method includes mathematical modeling, mathematical developments of statistical methods, writing of computer algorithms, and exemplification by reanalysis of published data sets from cancer and other medical studies and animal experiments with carcinogens. Models and methods developed in this research should lead to an improved understanding of event time data occurring in clinical research in cancer and other diseases--including the relation of event times to various risk factors, and quantification of group (familial) dependencies.